Noether Symmetries and Quantum Cosmology in Extended Teleparallel Gravity
NNoether Symmetries and Quantum Cosmology in Extended Teleparallel Gravity
Francesco Bajardi
1, 2, ∗ and Salvatore Capozziello
1, 2, 3, 4, † Department of Physics “E. Pancini”, University of Naples “Federico II”, Naples, Italy. INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Naples, Italy. Scuola Superiore Meridionale, Largo San Marcellino 10, I-80138, Naples, Italy. Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia. (Dated: February 2, 2021)We apply the Noether Symmetry Approach to point-like teleparallel Lagrangians in view to derive minisuper-spaces suitable for Quantum Cosmology. Adopting the Arnowitt–Deser–Misner formalism, we find out relatedWave Functions of the Universe. Specifically, by means of appropriate changes of variables suggested by theexistence of Noether symmetries, it is possible to obtain the cosmological Hamiltonians whose solutions areclassical trajectories interpretable as observable universes.
Keywords: Teleparallel gravity; quantum cosmology; Noether symmetries.
I. INTRODUCTION
General Relativity (GR) is considered the best accepted the-ory describing gravity. Despite its successes, related to the re-cent observational discovery of gravitational waves and blackholes, after more than one hundred years from its formula-tion, the theory presents many theoretical and experimentalissues, both at low and high energy scales, which need to beaddressed. At infrared scales, there are problems related tothe standard cosmological model and the lack of explanation,at fundamental scales, of dark energy and dark matter. At ul-traviolet scales, GR cannot be renormalized as the other fieldtheories, and in general its formalism cannot be adapted toQuantum Field Theory. Quantum theories of fields, in fact,deal with a standard Minkowskian background and fields canbe treated separately from spacetime. In GR, the field turnsout to be the background itself, and this represents a severeobstacle towards the construction of a self-consistent Quan-tum Gravity. The impossibility of fixing the ultraviolet diver-gences is also due to the fact that GR is a diffeomorphism-invariant, covariant theory which cannot be treated under thestandard Yang-Mills formalism on a fixed background.Considering this state of art, effective theories have beenproposed to cure GR shortcomings at large and microscopicscales. For example, at astrophysical and cosmological scales,extending the Einstein-Hilbert action through functions of theRicci scalar - f ( R ) theory - or through other curvature invari-ants might be a good starting point in order to address darkside problems, as shown e.g. in [1–8].On the other hand, in order to consider gravity under thesame standard as other interactions, it is necessary to pro-pose a quantization approach and describe gravitation by agauge formalism. One of the first quantization schemes wasintroduced by Arnowitt, Deser and Misner (ADM) which con-structed a formalism leading to a canonical quantization. Fur-ther details and applications of the ADM formalism can befound in [9–12]. This procedure is based on the concept of an ∗ [email protected] † [email protected] infinite-dimensional superspace constructed on the 3D- spa-tial metrics. Once a Super-Hamiltonian is defined on it, it ispossible to define a geometrodynamics to fix the evolution ofthese 3-metrics.The problem is extremely difficult to be handled from amathematical point of view, nevertheless, in 1983 J.B. Har-tle [13], using the Wentzel-Kramers-Brillouin approximation,showed that it is possible to restrict the superspace to finite-dimensional minisuperspaces where the SuperHamiltoniancan be quantized and a Wave Function of the Universe canbe analytically found as the solution of the Wheeler-De Witt(WdW) equation. In particular, Hartle formulated a criterionby which the Wave Function of the Universe, showing corre-lations and oscillating behaviors in the minisuperspace, givesrise to classical trajectories representing observable universes.The Wave Function of the Universe is only related to theprobability amplitude to obtain given configurations (see, e.g.[14]) but it does not give the full information on probability.The interpretation of the Wave Function has been discussedfor many years and it is still not completely clear. For in-stance, in its pioneering work, Everett proposed the so called Many Worlds Interpretation of Quantum Mechanics [15]. Inthis framework, the Wave Function of the Universe acquires aprobabilistic meaning. According to such an interpretation,all possible results of quantum measurements are simulta-neously realized in different universes without, therefore, aWave Function collapse. In other words, all the information isbrought by the Schrödinger equation (in this case by the WdWequation) and the wave function collapse giving real values ofthe observables (as in the Copenhagen picture) is not required.The weak point of the approach is understanding the role ofexperiments which would be able to verify the consistency ofmeasurements.Another interpretation was provided by Hawking, whostated that the Wave Function is related to the probability am-plitude for the early universe to develop towards our classicaluniverse.Despite these difficulties on the interpretation, the canon-ical quantization scheme is useful in Quantum Cosmologysince, thanks to the Wave Function, it is possible to recoverclassical trajectories representing observable universes. Inother words, the Quantum Cosmology application revealed a r X i v : . [ g r- q c ] F e b more useful than the full theory.However, the canonical quantization procedure does not al-low to obtain a full renormalizable theory of gravity to bedealt under the standard of Quantum Field Theory. A firststep aimed at solving such a problem, is to treat gravity as agauge theory. As pointed out in Sec. (II), it turns out thatthe gravitational interaction can be seen as a gauge theory ofthe local translation group and the corresponding Lagrangianturns out to be equivalent to that of GR up to a boundary term[16–19]. This theory is the so called Teleparallel Equivalentof General Relativity (TEGR) proposed by Einstein himselfsome years later the publication of GR [20]. According tosome authors, quantizing TEGR [21] should be a realistic ap-proach to realize a full Quantum Gravity theory.In this perspective, Quantum Cosmology related to TEGRand its extensions could be a useful exercise towards a com-prehensive quantum theory of gravity. This paper is devotedto this aim. In particular we want to show that quantizingminisuperspaces derived from TEGR and its extensions givesrealistic cosmological models. The main role in this study isplayed by the symmetries that, if exist for these minisuper-spaces, give a selection rule related to the Hartle criterion forclassical trajectories, as formerly demonstrated in [22, 23].In the next sections we briefly overview some basic as-pects of TEGR and its applications. Then we apply theNoether symmetry approach (see e.g. [23–33] ) to some ex-tended teleparallel Lagrangians containing functions of tor-sion scalar, teleparallel equivalent Gauss-Bonnet topologicalinvariant, and higher-order torsion terms. The purpose isto show that extended teleparallel models naturally exhibitNoether symmetries allowing a straightforward quantizationof the related minisuperspace models and then the possibilityto recover observable universes.The paper is organized as follows: in Sec. II, we discussthe main features and the basic foundations of TeleparallelGravity (for reviews on the topic, see [18, 20, 34]). In Secs.III, IV, V and VI, we apply the Noether approach to differ-ent cosmological point-like Lagrangians. Then, thanks to anappropriate change of variables suggested by Noether’s Theo-rem, we quantize the cosmological Hamiltonians and find thecorresponding Wave Function of the Universe, solution of theWdW-Schrödinger-like equation. In Sec. VII, we discuss re-sults and future perspectives.
II. TELEPARALLEL GRAVITY
Canonical quantization of GR, reported in Appendix A,does not fully address the problem of dealing with GR un-der the standard of Quantum Field Theory. Even assumingthe ADM formalism, it is not possible to treat GR as a uni-tary gauge field theory like Electroweak interaction or Quan-tum Chromo Dynamics. Several shortcomings arise in the at-tempt to quantize a tensor field which is also the backgroundof the theory. As said above, Quantum Field Theory instead,dealing with a fixed background, allows to handle a quantumformalism by imposing canonical commutation relations. Inthis picture, many ultraviolet divergences cancel out by means of Renormalization procedure. This latter, however, cannot beapplied to GR, whose divergences cannot be straightforwardlyregularized.Also for these reasons, it is possible to consider alternativetheories dealing with gravity as a gauge theory of the transla-tion group and describing the spacetime structure by torsioninstead of curvature (see [18] for a conceptual discussion onthis point). In what follows we recall some useful featuresrelated to TEGR, considering results reported in [35–39].Let us start by introducing the tetrad fields h aµ , used to lo-cally link the four-dimensional spacetime manifold with itstangent space, by means of the relation g µν = h aµ h bν η ab . (1)By means of the above relation, diffeomorphism transforma-tions can be thought as translations in the locally flat tangentspacetime. In this way, the covariant derivative of a genericvector field φ b can be written as: D a φ b = ∂ a φ b + ω bac φ c , (2)where ω is the spin connection. For a tensor field with mixedindexes, both the spin connection and the Levi-Civita connec-tion are involved, so that the covariant derivative of a rank-2tensor V aν turns out to be ∇ µ V aν = ∂ µ V aν + ω abµ V bν + Γ αµν V aα . (3)By requiring tetrads to satisfy the relation ∇ µ h aν = 0 and ne-glecting the spin connection, the so called Weitzenböck con-nection arises: Γ pµν = h pa ∂ µ h aν . (4)This assumption selects the class of frames with vanishingspin connection, though generally by means of "good tetrads"criteria it is possible to deal with both tetrads and spin connec-tions [40]. The connection (4) is not symmetric with respectto the lowest indexes and the antisymmetric part, that is T pµν := 2Γ p [ µν ] , (5)defines the torsion tensor . The contraction of the torsion ten-sor with a potential S µνp defines the torsion scalar : T = T pµν S pµν S pµν = K µνp − g pν T σµσ + g pµ T σνσ ,K νpµ = 12 (cid:0) T νp µ + T νµ p − T νpµ (cid:1) . (6)TEGR postulates the action S = (cid:90) h T d x , (7)to describe the gravitational interaction, where h is the de-terminant of the tetrad fields. It is possible to show that the Latin indexes label the tangent spacetime, while greek indexes are the co-ordinates labeling the standard spacetime. above action differs from the Einstein-Hilbert one only for aboundary term. Specifically, the relation R − h ∂ µ ( hT νµν ) = − T, (8)holds. This means that the action (7) automatically yields thesame field equations as GR. The field equations can be ob-tained by varying the action with respect to the tetrad fields,and read as ∂ σ ( h S pσa ) + h h λa S νpµ T µνλ = 0 . (9)Similarly to GR extensions, TEGR action can be modifiedwith the aim of solving the large-scale structure issues [18].The simplest extension is given by the action containing afunction of the torsion scalar, namely: S = (cid:90) hf ( T ) d x . (10)As showed in [41–43], some particular functions are able toexplain the cosmic accelerated expansion and the structureformation without introducing dark energy and dark matter.It is worth noticing that while teleparallel field equations areequivalent to those of GR, f ( R ) and f ( T ) theories are quitedifferent. For instance, as the former leads to fourth-orderequations of motion, the latter provides second-order equa-tions. This difference, besides simplifying the dynamics ofthe f ( T ) theory, has several implications in the cosmologicalframework. As an example, the further polarization modescarried by f ( R ) gravitational waves, vanish in the f ( T ) for-malism, whose modes are the same as in standard TEGR [44–48]. Extended TEGR field equations read: h ∂ µ ( h h pa S µνp ) f T ( T ) − h λa T pµλ S νµp f T ( T ) ++ h pa S µνp ( ∂ µ T ) f T T ( T ) + 14 h νa f ( T ) = 0; (11)In a Friedmann-Lemaître-Robertson-Walker (FLRW) spa-tially flat spacetime, the torsion scalar takes the form T = − (cid:18) ˙ aa (cid:19) , (12)and, unlike the cosmological expression of the Ricci scalar, itdoes not contain second derivatives. This permits to write thecosmological point-like Lagrangian of extended TEGR with-out integrating higher–order terms. III. f ( T ) COSMOLOGY
Let us start by applying the Noether Symmetry Approach to f ( T ) cosmology, the simplest extension of TEGR. The corre-sponding action is given by Eq. (10). In the FLRW universe,tetrad fields can be chosen as: h aµ = a a
00 0 0 a → h = a , (13) so that the correspondence (1) is respected. Notice that the di-agonal set of tetrads (13) is not the only one leading to a cos-mological spatially flat line element. In fact, while in GR themetric is uniquely determined once given the interval, in theteleparallel context, different tetrad fields can yield the sameline element.In order to find a suitable point-like Lagrangian, we use theLagrange multipliers method with the constraint given by Eq.(12), i.e. T = − H , with H being the Hubble parameter.In this way, all dynamical variables depend only on the cos-mic time and the three-dimensional surface term can be easilyintegrated. The action therefore reads: S = 2 π (cid:90) { a f ( T ) − λ ( T + 6 H ) } dt . (14)By varying the action with respect to the torsion, we get theform of the Lagrange Multiplier λλ = a ∂f ( T ) ∂T (15)and the point-like Lagrangian turns out to be L = a [ f ( T ) − T f (cid:48) ( T )] − a ˙ a f (cid:48) ( T ) , (16)where the prime denotes the derivative with respect to T . TheEuler-Lagrange equations and the energy condition for thetwo variables { a, T } are, respectively ddt ∂ L ∂ ˙ a = ∂ L ∂a ddt ∂ L ∂ ˙ T = ∂ L ∂T (17) EC → ˙ a ∂ L ∂ ˙ a + ˙ T ∂ L ∂ ˙ T − L = 0 , (18)and can be solved only after selecting the form of the function.Note that the Lagrangian is independent of ˙ T , so that the sec-ond Euler Lagrange equation provides the constraint imposedon the torsion scalar: a T + 6 ˙ a = 0 → T = − (cid:18) ˙ aa (cid:19) . (19)The Euler-Lagrange equation with respect to a and the energycondition lead to the following system of differential equa-tions: ¨ a + ˙ a a + ˙ a ˙ T f (cid:48)(cid:48) ( T ) f (cid:48) ( T ) − a T f (cid:48) ( T ) − f ( T ) f (cid:48) ( T ) = 0 a [ f ( T ) − T f (cid:48) ( T )] + 6 ˙ a f (cid:48) ( T ) = 0 . (20)Let us now apply the approach outlined in Appendix B tothe Lagrangian (16). It is worth noticing that the configura-tion space is Q ≡ { a, T } and the related tangent space is T Q ≡ { a, ˙ a, T, ˙ T } . According to the discussion in the In-troduction, Q is the minisuperspace on which we can developour Quantum Cosmology. A. Noether symmetries in f ( T ) cosmology Following Appendix B, the Noether vector in the minisu-perspace
Q ≡ { a, T } reads as X = α∂ a + β∂ T + ˙ α∂ ˙ a + ˙ β∂ ˙ T (21)with α = α ( a, T ) , β = β ( a, T ) being the components of theinfinitesimal generator η i .In order to select symmetries, we impose the vanishing Liederivative of the Lagrangian (16), that is L X L = 0 . (22)This procedure yields the following system of three partialdifferential equations: αf (cid:48) ( T ) + 2 af (cid:48) ( T )( ∂ a α ) + βaf (cid:48)(cid:48) ( T ) = 0 ∂ T α = 03 αa f ( T ) − αa T f (cid:48) ( T ) − βa T f (cid:48)(cid:48) ( T ) = 0 . (23)After some simple manipulations, from the above system it ispossible to get the infinitesimal generators α and β as well asthe form of the functions: α = α a − n β = − α n T a − n , (24) f ( T ) = f T n . (25)Therefore, f ( T ) = f T n is the only function admittingNoether symmetries [49]. Replacing into the equations of mo-tion (20), we have f ( T ) = f √ T , (26)in agreement with the results provided in Ref. [50]. Thismeans that the equations of motion further constrain the freeparameter n to n = 1 / . Notice that a linear combinationof the Euler-Lagrange equations gives the vacuum field equa-tions (cid:40)
12 ˙ a f T ( T ) + a f ( T ) = 012 ˙ a f T T ( T ) − a f T ( T ) = 0 . (27)Considering the cosmological expression of torsion, the aboveequations become (cid:40) T f T ( T ) − f ( T ) = 02 f T T ( T ) + f T ( T ) = 0 , (28)which are identically satisfied for f ( T ) = √ T .Let us now discuss the Hamiltonian formalism and the pos-sible applications to Quantum Cosmology. Finding the gener-ators α and β , with the help of eq. (B13), we can pass fromthe minisuperspace of the initial variable to the reduced spacecontaining a cyclic variable. This will allow an exact integra-tion of the field equations. B. The Hamiltonian formalism and the Wave Function of theUniverse
Relations (B13) allow to introduce a cyclic variable intothe system, passing from the minisperspace Q = { a, T } to Q (cid:48) = { z, w } with z being a cyclic variable, that is: (cid:40) α∂ a z + β∂ T z = 1 α∂ a w + β∂ T w = 0 . (29)By replacing the expressions of β and α in Eq. (29), a possiblesolution of the above system is w = a T n z = 2 n α a n , (30)or, equivalently a = (cid:18) α z n (cid:19) n T = w n (cid:18) α z n (cid:19) − , (31)so that the Lagrangian written in terms of the new variablesreads L = w (1 − n ) − nα ˙ z w n − n , (32)where z is a cyclic variable. It is worth noticing that, oncechanged the minisuperspace coordinates, the equations of mo-tion coming from Lagrangian (32) are simpler than those inEq. (17); they take the form: z → nw ¨ z + ˙ w ˙ z ( n −
1) = 0 (33) w → w = − n α n ˙ z n . (34)From Lagrangian (32), with a straightforward Legendre trans-formation, we get the Hamiltonian: H = w ( n −
1) + 3 π z nα w n − n . (35)The canonical quantization procedure can be pursued by re-placing the operator i∂ z to the momentum π z and, by applyingthe relation (A13) to the Hamiltonian (35), we get the system (cid:40) H ψ = 0 i∂ z ψ = Σ ψ (36)where the first is Wheeler-DeWitt equation and the second isthe conserved momentum. Explicitly, we have (cid:34) w ( n − − ∂ z nα w n − n (cid:35) ψ = 0 i∂ z ψ = Σ ψ . (37)The solution of the second equation is ψ = ψ e i Σ z , (38)which, combined with the first, yields the Wave Function ofthe Universe, that is: ψ ∼ exp (cid:110) i (cid:104) α w n − n (cid:112) n ( n − (cid:105) z (cid:111) . (39)This wave function is oscillating so, according to the Hartlecriterion, permits to select classical trajectories (see [22] for adetailed discussion). By recasting ψ in terms of the action S as ψ ∼ exp { iS } , (40)and identifying the action with the quantity S = (cid:104) α w n − n (cid:112) n ( n − (cid:105) z, the Hamilton-Jacobi equations give: ∂S ∂z = π z = Σ ∂S ∂w = π w = 0 . (41)Note that the above equations are nothing but the Euler-Lagrange equations for the new variables defined in Eqs. (33)and (34). The general solution is z ( t ) = z t w = − n α n z n . (42)Coming back to the old variables, the scale factor and the tor-sion scalar can be written as functions of time a ( t ) = a t n T ( t ) = − n t (43)for any f ( T ) = T n except for n = 1 / . Once merged withthe energy condition ( ˙ z∂ ˙ z + ˙ w∂ ˙ w − L = 0 , the equationsare further constrained to the subcase n = 1 / , which turnsout to be a trivial case, as pointed out at the beginning of thissection. In summary, the existence of the Noether symme-try allowed to integrate exactly the dynamical system and,furthermore, it allows to find out classical trajectories in theminisuperspace which can be interpreted as observable uni-verse. In the specific case of (43), depending on the valueof n , we can achieve Friedmann decelerating solutions (for < n < / or power-law inflationary solutions. IV. F ( T, (cid:3) T ) COSMOLOGY
A further extension of TEGR can come considering higher-order derivative terms in the torsion scalar as (cid:3) T , being (cid:3) the d’Alembert operator (cid:3) = D µ D µ [48]. It represents acase of particular interest, since the corresponding metric the-ory F ( R, (cid:3) R ) is renormalizable at one-loop and higher-loop level [27, 51, 52]. In general, higher-order terms arise by con-sidering quantum corrections to GR, as shown in [53, 54] andthen it is worth studying their effects also in TEGR.In principle, the expression of the operator (cid:3) is differentfrom the one in GR, due to the different form of connections.In this case, the Weitzenböck connection introduced in Sec. IIyields: (cid:3) T = ∇ µ ∇ µ T = ∇ µ ∂ µ T = ∂ µ ∂ µ T + Γ pµp ∂ µ T == ¨ T + Γ p p ˙ T = ¨ T + 3 (cid:18) ˙ aa (cid:19) ˙ T . (44)Note that despite the different form of the connection, thed’Alembert operator has the same expression as in GR.Thanks to Eq. (44), we can use the Lagrange multipliersmethod in order to get the point-like Lagrangian; to developthe approach, we treat the torsion T and its d’Alembertian (cid:3) T as separated fields, although they are related through Eq.(44).Let us start by writing the general action S = (cid:90) hF ( T, (cid:3) T ) d x (45)so that, after integrating the three-dimensional hyper-surface,considering the Lagrange multiplier gives: S = 2 π (cid:90) (cid:26) a F ( T, (cid:3) T ) − λ (cid:18) T + 6 ˙ a a (cid:19) + − λ (cid:18) (cid:3) T − ¨ T − aa ˙ T (cid:19)(cid:27) dt . (46)with λ and λ being the two Lagrange multipliers. By vary-ing the action with respect to T and (cid:3) T we find: λ = a ∂F ( T, (cid:3) T ) ∂T (47)and λ = a ∂F ( T, (cid:3) T ) ∂ (cid:3) T . (48)Replacing Eqs.(47) and (48) into the action (46) and integrat-ing the second-order time derivatives, we finally get L = a (cid:18) F − T ∂F∂T − (cid:3) T ∂F∂ (cid:3) T (cid:19) + − a ˙ a ∂F∂T − a ˙ T ˙ (cid:3) T ∂ F∂ (cid:3) T − a ˙ T ∂ F∂T ∂ (cid:3)
T . (49)Note that by setting ∂F∂ (cid:3) T = 0 we recover the Lagrangian(16), discussed in Sec. III. The Euler-Lagrange equation withrespect to (cid:3) T provides the cosmological expressions of T and (cid:3) T , while the Euler-Lagrange equation related to thetorsion gives the constraint (cid:3) f (cid:3) T = 0 . The equation withrespect to the scale factor, together with the energy condi-tion, provide the dynamics of the variables in the minisu-perspace Q ≡ { a, T, (cid:3) T } whose tangent space is T Q ≡ { a, ˙ a, T, ˙ T , (cid:3) T, ˙ (cid:3) T } . The complete system of differentialequations reads as a ¨ aF T ( T, (cid:3) T ) + 4 a ˙ a ˙ (cid:3) T F T (cid:3) T ( T, (cid:3) T ) ++4 a ˙ a ˙ T F
T T ( T, (cid:3) T ) + 2 ˙ a F T ( T, (cid:3) T ) + − a ˙ (cid:3) T ˙ T F (cid:3) T (cid:3) T ( T, (cid:3) T ) − a ˙ T F T (cid:3) T ( T, (cid:3) T ) + − a (cid:3) T F (cid:3) T ( T, (cid:3) T ) − a T F T ( T, (cid:3) T ) ++ a F ( T, (cid:3) T ) = 0 (cid:3) f (cid:3) T = 0 ∂ F ( T, (cid:3) T ) ∂ (cid:3) T (cid:18) (cid:3) T − aa ˙ T − ¨ T (cid:19) ++ ∂ F ( T, (cid:3) T ) ∂T ∂ (cid:3) T (cid:18) T + 6 ˙ a a (cid:19) = 06 ˙ a F T ( T, (cid:3) T ) + a ˙ (cid:3) T ˙ T F (cid:3) T (cid:3) T ( T, (cid:3) T ) ++ a ˙ T F T (cid:3) T ( T, (cid:3) T ) − a (cid:3) T F (cid:3) T ( T, (cid:3) T ) + − a T F T ( T, (cid:3) T ) + a F ( T, (cid:3) T ) = 0 . (50)As before, the above equations cannot be solved until thefunction F ( T, (cid:3) T ) is selected. A. Noether symmetries for F ( T, (cid:3) T ) cosmology The application of the condition L X L = 0 to the La-grangian (49) allows to find the symmetries of the high-ordertheory F ( T, (cid:3) T ) . The minisuperspace of configurations con-tains three variables, so that Noether’s vector has the form: X = α∂ a + β∂ T + γ∂ (cid:3) T + ˙ α∂ ˙ a + ˙ β∂ ˙ T + ˙ γ∂ ˙ (cid:3) T . (51)With respect to the previous section, the theory contains a newinfinitesimal generator γ , related to the presence of the newvariable (cid:3) T . Imposing the condition X L = 0 and equatingto zero the coefficients of ˙ a , ˙ T ˙ (cid:3) T , ˙ T , ˙ a ˙ T , ˙ a ˙ (cid:3) T and ˙ (cid:3) T , we get a system of seven differential equations: α (cid:18) F − T ∂F∂T − (cid:3) T ∂F∂ (cid:3) T (cid:19) − βaT ∂ F∂T + − βa (cid:3) T ∂ F∂T ∂ (cid:3) T − γaT ∂ F∂T ∂ (cid:3) T − γa (cid:3) T ∂ F∂ (cid:3) T = 0 (cid:18) α + 2 a ∂α∂a (cid:19) ∂F∂T + βa ∂ F∂T ++ γa ∂ F∂T ∂ (cid:3) T = 0 (cid:18) α + a ∂γ∂ (cid:3) T + a ∂β∂T (cid:19) ∂ F∂ (cid:3) T + βa ∂ F∂T ∂ (cid:3) T ++ γa ∂ F∂ (cid:3) T = 0 (cid:18) α + 2 a ∂β∂T (cid:19) ∂ F∂T ∂ (cid:3) T + βa ∂ F∂T ∂ (cid:3) T ++ γa ∂ F∂T ∂ (cid:3) T + a ∂γ∂T ∂ F∂ (cid:3) T = 012 ∂α∂T ∂F∂T + a ∂γ∂a ∂ F∂ (cid:3) T + 2 a ∂β∂a ∂ F∂T ∂ (cid:3) T = 012 ∂α∂T ∂F∂T + a ∂β∂a ∂ F∂ (cid:3) T = 0 ∂β∂ (cid:3) T = 0 , (52)whose non trivial solutions are α = 0 β = β γ = β (cid:3) TTF I ( T, (cid:3) T ) = F T − k (cid:3) T k α = − γ k (cid:3) T a − β = β a − γ = γ ( (cid:3) T ) a − F II ( T, (cid:3) T ) = f ( (cid:3) T ) k ++ f T α = − γ ka (cid:3) T χ ( a ) β = β χ ( a ) γ = γ ( (cid:3) T ) F III ( T, (cid:3) T ) = f ( (cid:3) T ) k α = α a − k β = − α k T a − k γ = γ ( a, T, (cid:3) T ) F IV ( T, (cid:3) T ) = f T k + f (cid:3) T. (53)The first function is the only one made of a product of T and (cid:3) T ; it is of particular interest, since the product between thetwo variables can be seen as a modified f ( T ) theory of grav-ity non-minimally coupled to a scalar field. The infinitesi-mal generators of the second solution contain a function ofthe scale factor; the corresponding function consists in a sumof the two variables T and (cid:3) T , as well as the fourth func-tion. Specifically, this latter leads to the same Lagrangian as f ( T ) gravity in Eq. (16). This means that the additive con-tribution of the higher-order term (cid:3) T does not introduce anyfurther degrees of freedom; in such a case, also the generatorcoefficients are, in turn, the same as those found in Sec. III.In what follows, we focus on the first solution of Noethersystem, discussing the Hamiltonian dynamics and solving theWheeler-DeWitt equations in the minisuperspace of the threevariables a, T, (cid:3) T . The point-like Lagrangian can be foundby replacing the function F I ( T, (cid:3) T ) into Eq. (49), obtaining L = F ( k − (cid:40) a ˙ a (cid:18) (cid:3) TT (cid:19) k − ka ˙ T ˙ (cid:3) T (cid:18) (cid:3) TT (cid:19) k − T ++ ka ˙ T (cid:18) (cid:3) TT (cid:19) k − T (cid:41) . (54)The Euler-Lagrange equations can be analytically solved for k ≥ providing a de Sitter-like cosmological solution of theform a ( t ) = a e nt , T ( t ) = − n , (cid:3) T ( t ) = 0 , (55)with n being an arbitrary real number. To find the Hamiltonianand the corresponding wave function, we use the conditionin Eq. (B13), by means of which the minisuperspace Q ≡{ a, T, (cid:3) T } can be transformed to Q (cid:48) ≡ { z, ω, u } , where z isa cyclic variable. The change of variables yields the system α ∂z∂a + β ∂z∂T + γ ∂z∂ (cid:3) T = 1 α ∂ω∂a + β ∂ω∂T + γ ∂ω∂ (cid:3) T = 0 α ∂u∂a + β ∂u∂T + γ ∂u∂ (cid:3) T = 0 . (56)whose possible solution is z = z ( T ) = Tβ , ω = (cid:3) TT , u = a , (57)from which the variables T and (cid:3) T can be written in terms ofthe new variables ω and z as: T = β z, (cid:3) T = β ωz, a = u (58)The Lagrangian L ( z, ω, u ) turns out to be L new = F ( k − (cid:8) a ˙ a ω k − kβ a ω k − ˙ z ˙ ω (cid:9) . (59)As expected, the shape of the Lagrangian suggests that z isnow a cyclic variable, so that the constant of motion Σ is theconjugate momentum related to the cyclic variable: Σ = ∂ L ∂ ˙ z = − F β k ( k − a ω k − ˙ ω . (60) B. The Hamiltonian and Wave Function of the Universe
For the sake of simplicity, we define F k ( k − β = p and F ( k −
1) = p , so that the conjugate momenta can be Since u = a we prefer to write Lagrangian depending on a instead of u . written as: π z = ∂ L ∂ ˙ z = − p a ω k − ˙ ωπ ω = − p a ω k − ˙ zπ a = p aω k ˙ a (61)and, finally, the Hamiltonian reads H = − p a ω k − (cid:0) π z + π ω − π z π ω (cid:1) − π a p aω k . (62)The constraints imposed by Eq. (A9) provide the system ofdifferential equations (cid:0) ∂ z + ∂ ω + ∂ z ∂ ω (cid:1) ψ + 12 p p a ω ∂ a ψ = 0 − i∂ z ψ = Σ ψ . (63)The solution of the former equation is a linear combination ofBessel functions, whose asymptotic limit provides an oscillat-ing wave function of the form ψ ∼ exp (cid:40) i (cid:34)
12 ln ω − √
32 Σ ω − π (cid:0) √ c + 1 (cid:1) + − Σ z + √ c ln a ] } (64)Notice that the Hartle criterion is then recovered for largescale factors, where the wave function is peaked in the min-isuperspace Q (cid:48) . Moreover, classical trajectories can be recov-ered by identifying the exponential factor of Eq. (64) with S ,namely: S = 12 ln ω − √
32 Σ ω − π (cid:0) √ c + 1 (cid:1) − Σ z + √ c ln a (65)and the wave function can be recast as ψ ∼ e iS . TheHamilton-Jacobi equations coming from the action (65) readas ∂S ∂a = π a ∂S ∂z = π z ∂S ∂ω = π ω , (66)with π a = √ c a π ω = 12 ω − √
32 Σ π z = Σ , (67)from which a system of three differential equations follows: p aω k ˙ a = √ c ap a ω k − ˙ z = √
32 Σ − ωp a ω k − ˙ ω = Σ . . (68)The solutions of the above system are the same as Eq. (55), sothat classical trajectories, and then observable universes, areimmediately recovered. V. TELEPARALLEL GAUSS-BONNET COSMOLOGY
Let us consider now the contribution of the Gauss-Bonnettopological invariant among the possible TEGR extensions. Infour-dimensional metric formalism, the Gauss-Bonnet scalaris a topological invariant which assumes the form: G = R − R µν R µν + R µνpσ R µνpσ . (69)In FLRW cosmology, the square root of the Gauss-Bonnetscalar is dynamically equivalent to the Ricci scalar, since thecontribution of other second order curvature invariants arecomparable with respect to that of R . Being a topologicalsurface term, in four dimensions, the action containing R + G provides the same dynamics as the Einstein-Hilbert action;however, in five dimensions or more the integral of G is nota topological invariant, so that the action S = (cid:82) ( R + G ) d n x (with ≥ ) does not yield the same equations of motion as the n -dimensional GR. Another issue which may be solved by in-troducing G in the theory, is linked to the treatment of gravityunder the gauge formalism, since the Gauss-Bonnet invariantnaturally emerges in many gauge theories as Chern-Simonsor Lovelock gravity. The Gauss-Bonnet term is consideredas a part of the gravitational action in several works, such as[55, 56] where a function of R and G is discussed, or [57, 58]where GR is recovered from f ( G ) gravity without imposingthe Einstein-Hilbert action in R a priori .Moreover, it is possible to construct a teleparallel equiva-lent Gauss-Bonnet term as shown in [59–61].The aim of this section is to apply the formalism of Quan-tum Cosmology to a function of the torsion and of the telepar-allel Gauss-Bonnet invariant, finding the Wave Function ofthe Universe and recovering classical trajectories related toobservable universes.The cosmological expression of the teleparallel Gauss-Bonnet term is equivalent to that provided by the correspond-ing metric theory [62], namely: G = T G = 24 ¨ a ˙ a a , (70)from which it is easy to verify that it represents a total deriva-tive. Considering that a function of T G is not-trivial even in afour-dimensional spacetime, we take into account an action ofthe form S = (cid:90) | e | f ( T, T G ) d x, (71)so that TEGR is recovered as soon as f ( T, T G ) = T . Bymeans of Noether’s approach, it is possible to find out thecosmological solutions for a generic function of the telepar-allel surface term T G containing symmetries; this approachhas been performed in [61], therefore in this section we onlyoutline the main results. After this, we consider the ADM for-malism and find the related Wave Function of the Universe,as we did before. In the minisuperspace Q ≡ { a, T, T G } , thesymmetry generator is X = α ( a, T, T G ) ∂ a + β ( a, T, T G ) ∂ T + γ ( a, T, T G ) ∂ T G . (72) The application of the condition L X L = 0 to the point-likeLagrangian L = a ( f − T G f T G − T f T ) − a (cid:16) T G f T G T G + ˙ T f
T T G (cid:17) − f T a ˙ a , (73)gives rise to the following Noether solutions [61]: X = (cid:26) β ( a, T G , T ); TT G β ( a, T G , T ) (cid:27) , f ( T, T G ) = f T k G T − k (74)and then the point-like Lagrangian reduces to L = f ( k −
1) ˙ a T k − G T − k (cid:104) k ˙ a ( T G ˙ T − T ˙ T G ) + 3 aT G (cid:105) . (75)The solutions of the Euler-Lagrange equations are a ( t ) = a t k +1 , T = − k + 1) t , T G ( t ) = 24 2 k (2 k + 1) t . (76)Here k is an arbitrary real number and then the above con-siderations work, i.e. it is possible to recover Friedmann andinflationary cosmological solutions.In order to find the Hamiltonian, we perform the change ofvariables by means of which we can introduce a cyclic vari-able z ; the system (B13) can be written as: β∂ T G z + β TT G ∂ T z = 1 ∂ T G w + TT G ∂ T w = 0 ∂ T G u + TT G ∂ T u = 0 (77)and, by assuming β = β T G , one possible solution is z = 1 β ln( T G ) w = T G T u = a , (78)from which a = u T G = e β z T = e β z w . (79)Thanks to these relations, Lagrangian (75) takes the form: L = f ( k −
1) ˙ a w k (cid:20) − k ˙ a ˙ ww + 3 a (cid:21) . (80)Notice that the cyclic variable z does not appear in the La-grangian and the minisuperspace is further restricted to thetwo variables w and a . By means of the conserved quantity Σ (cid:18) − w − k π w f k ( k − (cid:19) = Σ , (81)the Hamiltonian can be written as: H = Σ π a − f k ( k − aw k Σ . (82)From the canonical quantization rules, after writing Hamilto-nian in terms of operators, the Wheeler-De Witt equation takesthe form ˆ H ψ = − i∂ a ψ − f ( k − aw k Σ ψ = 0 , (83)whose solution is: ψ = ψ exp (cid:26) i (cid:20)
32 Σ f ( k − w k (cid:21) a (cid:27) . (84)The Hartle criterion is recovered and the Hamilton-Jacobiequations provide the above classical trajectories. VI. NON-MINIMALLY COUPLED SCALAR FIELD
The final case we are going to analyze is teleparallel scalar-tensor TEGR, whose action reads as: S = (cid:90) (cid:104) T F ( φ ) + ω ∂ µ φ∂ µ φ − V ( φ ) (cid:105) d x (85)being φ a scalar field, F ( φ ) the non-minimal coupling, and V ( φ ) the self-interacting potential. We can also take into ac-count more general actions, depending on a general function F ( T, φ ), however, here we restrict to the case linear in T . Theabove action is the teleparallel equivalent of the well knownscalar-tensor action in the metric formalism (see [63, 64]) dis-cussed e.g. in [65, 66]. In FLRW cosmology, where the hyper-surface term can be integrated, we can recast the action as anintegral over the time where variables are only time depen-dent, namely S = 2 π (cid:90) h (cid:20) F ( φ ) T + 12 ˙ φ − V ( φ ) (cid:21) dt , (86)Replacing therefore the cosmological form of the torsion intothe action, the Lagrangian reads: L = L ( a, ˙ a, φ, ˙ φ ) = − F ( φ ) a ˙ a + (cid:20)
12 ˙ φ − V ( φ ) (cid:21) a . (87)In this case, the minisuperspace only consists of the two vari-ables a and φ , that is Q ≡ { a, φ } ; the torsion does not ap-pear manifestly as a variable, having been replaced by its cos-mological expression. Starting from the Lagrangian (87), theequations of motion and the energy condition E L = 0 yieldthe system of differential equations: a ˙ a ˙ φF φ ( φ ) + 4 ˙ a F ( φ ) + 4 a ¨ aF ( φ ) + 12 a ˙ φ − a V ( φ ) = 03 a ˙ a ˙ φ + a ¨ φ + 6 ˙ a F φ ( φ ) + a V φ ( φ ) = 06 F ( φ ) ˙ a − a ˙ φ − a V ( φ ) = 0 . (88)The system can be solved after selecting the form of the cou-pling and the potential by Noether’s symmetries. A. Noether symmetries
Considering the Noether vector of the two-dimensionalminisuperspace Q = { a, φ } , namely X = α∂ a + β∂ φ + ˙ α∂ ˙ a + ˙ β∂ ˙ φ (89)and setting the Lie derivative of the Lagrangian (87) along X equal to zero, gives the following system of partial differentialequations: αF ( φ ) + βaF (cid:48) ( φ ) + 2 F ( φ ) a∂ a α = 03 α + 2 a∂ φ β = 0 a ∂ a β − F ( φ ) ∂ φ α = 03 αV ( φ ) + βaV (cid:48) ( φ ) = 0 , . (90)whose non trivial solutions are α = − β (cid:112) V aβ = β (cid:112) V φV I ( φ ) = V φ F I ( φ ) = F φ α = − β (cid:96) + 3 a (cid:96) +1 φ − (cid:96) (cid:96) +3 β = β a (cid:96) φ (cid:96) +3 V II ( φ ) = V φ (cid:96) +3 F II ( φ ) = (2 (cid:96) + 3) φ ; α = 2 qβ e qφ √ aβ = β e qφ √ a V III ( φ ) = V e qφ F III ( φ ) = 316 q α = − a ( c + 2 c φ ) β = a − ( c + c φ + c φ ) V IV ( φ ) = V ( c + c φ + c φ ) F IV ( φ ) = 364 c ( c + c φ + c φ ) (91)Let us focus on the first one, with the aim to investigate boththe classical and the quantum cosmological implications. Thepoint-like Lagrangian, which arises once replacing the func-tions F I ( φ ) and V I ( φ ) into Eq. (87), takes the form L = − F aφ ˙ a + a (cid:20)
12 ˙ φ − V φ (cid:21) , (92)so that the conserved quantity is Σ = α∂ a L + β∂ φ L = β (cid:112) V a φ (8 F φ ˙ a + a ˙ φ ) . (93)The equations of motion (88), when F (cid:54) = 0 , V (cid:54) = 0 , onlyprovide exponential solutions of the form a ( t ) = a e kt φ = φ e F k ( t − t ) V ( φ ) = 2 k F (3 − F ) φ F ( φ ) = F φ . (94)By neglecting the contribution of the potential, the Euler-Lagrange equations provide also power-law solutions, whichfurther constrain F ( φ ) to a ( t ) = a t k φ ( t ) = φ t (1 − k ) F ( φ ) = (3 k − k φ . (95)0From the above power-law expressions of the scale factor, wecan distinguish the three different Friedmann erasRadiation Dominated Era : → a ( t ) = a t φ ( t ) = φ t − Stiff Matter Dominated Era : → a ( t ) = a t φ ( t ) = φ Dust Matter Dominated Era : → a ( t ) = a t φ ( t ) = φ t − The evolution of the scalar field can be summarized in thefigure below:Fig. 1:
Behavior of the scalar field in the three cosmologicaleras as functions of time.
We now apply the relations in Eqs. (29) to find the cyclicvariable z in the minisuperspace. It yields the system − a∂ a z + φ∂ φ z = 1 − a∂ a w + φ∂ φ w = 0 . (96)Assuming w = a φ and ∂ a z = 0 , one of the possible solu-tion is (cid:40) z = ln φw = a φ (cid:40) φ = e z a = w e − z (97)and the new Lagrangian is then L new = − F ˙ w w + 83 F ˙ w ˙ z + f w ˙ z − V w , (98)where we set f ≡ − / F . Here z is the cyclic vari-able and the conserved quantity can be written in terms of themomentum π z as: ∂ ˙ z L = π z = 83 F ˙ w + 2 f w ˙ z = Σ . (99) B. Hamiltonian and Wave Function of the Universe
The ADM formalism can be pursued by writing the time-derivatives of the cosmological variables as functions of mo-menta and performing a Legendre transformation; this proce-dure yields the Hamiltonian : H = π z w − f F wπ w + 2 π z π w + V w . (100) The Wave Function of the Universe is the solution of the sys-tem − w ∂ z ψ ( z, w ) − ∂ w ∂ z ψ ( z, w )++ 12 f F w∂ w ψ ( z, w ) + V wψ ( z, w ) = 0 i∂ z ψ ( z, w ) = Σ ψ ( z, w ) . (101)The first equation is the Wheeler-De Witt equation, comingfrom ˆ H ψ = 0 ; the second equation is conservation relationprovided by the Noether symmetry, namely π z = Σ . Thesolution of (101) is a linear combination of Bessel’s functions,which, for large arguments, are peaked in the minisuperspacevariables z, w . Therefore the wave function has an asymptoticbehavior of the form ψ ( z, w ) ∼ e i ( Σ z − ln w + W w − p π − π ) , (102)where p , A , B and W are defined as p = 12 (cid:112) − A − iA − B + 1 − F f = A ; 2 F f V = W ; F f Σ = B. (103)Notice that Hartle’s criterion is recovered after imposing thechange of variables suggested by the Noether symmetry; fur-thermore, in the semiclassical limit where the wave functioncan be recast as ψ ∼ e iS , the Hamilton-Jacobi equationsyield a system of two differential equations of the form (cid:40) π z = ∂ z S = Σ π w = ∂ w S = W − w , (104)whose solution is a ( t ) ∼ e kt φ ( t ) ∼ e (cid:96)t . (105)Eqs. (104) provide the same exponential solution as the Euler-Lagrange equations, as expected by construction. The power-law solution (95) can be recovered for the potential V ( φ ) =0 . Notice that even in this case Noether symmetry allows tocalculate the equations of motion solutions by means of thesemiclassical limit of the ADM formalism. VII. DISCUSSION AND CONCLUSIONS
The purpose of this paper is to apply the Noether Symme-tries Approach to different extended TEGR models in view ofQuantum Cosmology applications. In several works [67–70],the Noether Approach has been used to select the Lagrangiansdepending on functions of curvature invariants, using the sym-metries and reducing therefore the dynamics. In this paperwe found out exact solutions coming from actions contain-ing functions of torsion scalar T and its related invariants. Inall cases, we dealt with the quantum counterpart by using theADM formalism.1TEGR models have been proposed with the aim to relax thestrict constraints provided by GR, as well as the strict depen-dence on Equivalence Principle, metricity or Lorentz invari-ance; another assumption of GR concerns the Levi-Civita con-nection supposed to be torsionless. Relaxing the hypothesis oftorsionless connection, it is possible to construct spacetimeswhere affinities have a dynamical role, instead of geodesicstructure. TEGR is completely equivalent to GR at the levelof equations and then of dynamics. This can be easily under-stood by the fact that TEGR and GR field equations differ onlyfor a 4-divergence. However, though TEGR provides the sameresults as GR, extending TEGR turns out to be different withrespect to GR extensions, since this latter leads to higher-orderfield equations with respect to the metric. Nevertheless, f ( R ) gravity can be always obtained from f ( T ) gravity consideringthe boundary term B : a function f ( T, B ) , under appropriateconstraints, allows to recover the f ( R ) gravity [48, 71–73].One of the main advantages of studying TEGR and its ex-tensions is that these models can be easily recast as gaugetheories allowing deep insights in the fundamental structureof gravity. In this perspective, quantizing TEGR and exten-sions could be extremely interesting towards the realization ofQuantum Gravity. Therefore, Quantum Cosmology could beuseful to achieve results possibly comparable with observa-tions.Specifically, we worked out minisuperspace models by theNoether Symmetry Approach. We have also showed, byadopting suitable Lagrange multipliers, that Noether symme-tries provide constraints on the form of the action that allow tosimplify the dynamics and to find classical solutions. In par-ticular, the Noether approach allows to select changes of vari-ables such that the Wave Function of the Universe turns out tobe peaked in minisuperspaces containing cyclic variables. Inthis case, classical trajectories, representing observable uni-verses can be recovered. However, it is worth remarking thatneither Noether system solutions nor the change of variablesare unique. This means that a careful choice is often importantto recover exact useful solutions capable of describing classi-cal universes.As the Wave Function of the Universe is related to the prob-ability to get observable universes, the existence of Noethersymmetries suggests when the Hartle criterion is working.Due to the experimental lack of many copies of the same uni-verse, the resulting Wave Function is only related to the prob-ability to get a certain configuration, but it does not give thewhole probability amplitude itself. This shortcoming is alsodue to the lack of a self-consistent theory of Quantum Gravity.In particular, we focused on f ( T ) , F ( T, (cid:3) T ) , f ( T, G ) andon non-minimal coupled teleparallel models. The applicationof Noether Symmetry Approach to the related point-like La-grangians allows to select the functional form of the models.In all cases, the existence of symmetries permits to integratethe corresponding dynamical systems and to apply the Hartlecriterion to find out observable universes.In a forthcoming paper, the method will be systematicallyconfronted with observations to link the cosmological param-eters with the Noether symmetries. ACKNOWLEDGMENTS
The Authors acknowledge the support of
Istituto Nazionaledi Fisica Nucleare (INFN) ( iniziative specifiche
GINGER,MOONLIGHT2, QGSKY, and TEONGRAV). This paperis based upon work from COST action CA15117 (CAN-TATA), COST Action CA16104 (GWverse), and COST actionCA18108 (QG-MM), supported by COST (European Cooper-ation in Science and Technology).
Appendix A: The Arnowitt-Dese- Misner formalism andWheeler-DeWitt equation
In order to understand the early time behavior of the Uni-verse, we should treat it by using Quantum Field Theory, andrepresent its evolution by means of a Wave Function depend-ing on the space-time variables. This is not fully possible dueto the lack of a self-consistent theory of Quantum Gravity.However, adopting a canonical quantization approach andthe related ADM formalism allows to obtain the so called
Wheeler-DeWitt equation, which is a Schroedinger-like equa-tion whose solution, the Wave Function of the Universe, is auseful indication of probability amplitude for quantized super-space variables.In order to build up the formalism, let us start by consid-ering the most general form of the Einstein-Hilbert action,namely [74, 75] S = 12 (cid:90) V √− g [ R − λ ] d x + (cid:90) ∂V √ hK dx , (A1)where K = h ij K ij is the trace of the extrinsic curvature ten-sor of the three-dimensional hyper-surface ∂V surrounded inthe four-dimensional manifold, and h is the determinant ofthe three-dimensional metric . To get the Hamiltonian formu-lation, we perform a (3 + 1) -decomposition of the metric g µν ,so that the spatial components turn out to be the dynamicaldegrees of freedom. Choosing a set of coordinates X α , bymeans of coordinates foliation transformation X α → X (cid:48) α ,we can get a family of hypersurfaces x = const where x i are the local coordinates on each surface.In the four dimensions, to each time-like point x , it corre-sponds a space-like hypersurface x = k , and the variation of x provides the required foliation. Moreover, any point of thehypersurface is labeled by a three-dimensional vectors basis X αi , tangent to the surface itself. Since they are orthonormalto the surface vector n ν , the following two conditions hold: g µν X µi n ν = 0 g µν n µ n ν = − . (A2)It is also useful to define the deformation tensor as N α = ˙ X α = ∂ X α ( x , x i ) (A3) In this appendix, h ij is the spatial component of the spacetime metric g µν N α in the previously defined basis ofnormal and tangent vectors as N α = N n α + N i X αi , (A4)the metric tensor takes the form: g µν = (cid:18) − ( N − N i N i ) N j N j h ij . (cid:19) (A5)The scalar N and the vector N i are the so-called Lapse and
Shift functions, respectively. With respect to these decompo-sitions, the Lagrangian density turns out to be L = 12 √ hN (cid:16) K ij K ij − K + (3) R (cid:17) + T.D. (A6)By defining the conjugate momenta as π ≡ δ L δ ˙ N = 0 π i ≡ δ L δ ˙ N i = 0 π ij ≡ δ L δ ˙ h ij = √ h (cid:0) Kh ij − K ij (cid:1) (A7)and the Hamiltonian density as H = π ij ˙ h ij − L , (A8)the commutation relations between the Hamiltonian H and theconjugate momenta π are [23]: ˙ π = −{H , π } = δ H δN = 0˙ π i = −{H , π i } = δ H δN i = 0 . (A9)With these definitions in mind, the first step for the canonincalquantization is to transform classical dynamical variables tooperators by the appropriate commutation relations: [ˆ h ij ( x ) , ˆ π kl ( x (cid:48) )] = i δ klij δ ( x − x (cid:48) ) δ klij = ( δ ki δ lj + δ li δ kj )[ˆ h ij , ˆ h kl ] = 0[ˆ π ij , ˆ π kl ] = 0 , (A10)Notice that the conjugate momenta are now promoted to op-erators, so that the definitions in Eq. (A7) take the form ˆ π = − i δδN ˆ π i = − i δδN i ˆ π ij = − i δδh ij (A11)Furthermore, from Eq. (A9), it automatically follows that thecanonical quantization imposes the constraint ˆ H| ψ > = 0 (A12)which leads to the Wheeler-De Witt equation, namely aSchrödinger-like equation of the form: (cid:18) ∇ − √ h (3) R (cid:19) | ψ > = 0 . (A13) The function ψ is defined on the configuration space of the3-metrics, with ψ ≡ ψ [ h ij ( x )] describing the evolution of thegravitational field. The operator ∇ , is defined through the3-dimensional metric as ∇ = 1 √ h ( h ik h jl + h il h jk − h ij h kl ) δδh ij δδh kl . (A14)The space on which the Wave Function is defined is thespace of all the possible 3-metrics, called superspace . Sincethe Superspace is an infinite dimensional space, in order tosolve the Wheeler-De Witt equation, restrictions to finite-dimensional minisuperspaces are often needed. In such a way,the Wheeler-De Witt equation becomes a partial differentialequation. It is worth noticing that minisuperspaces represents toy models which preserves some basic aspects of the entiretheory, as some symmetry, but neglect others. From this pointof view, the strongest hypothesis of Quantum Cosmology is toassume that these models are workable approximations of thecomplete theory.However, some difficulties in the interpretation of the WaveFunction of the Universe arises: in non-relativistic Quan-tum Mechanics, the dynamics of the system is described bythe Schrödinger equation, so that the quantity < ψ | ψ > ≡ (cid:82) ψ ∗ ψ dx is always positive and independent of time. Theconsequence of this fact is that solutions of the Schrödingerequation constitute an infinite dimensional space, and theproduct ψ ∗ ψ can be interpreted as the probability density tolocalize the particle into a give configuration space. On thecontrary, the Wheeler-DeWitt equation is similar to the KleinGordon equation: it is possible to find a continuity equation,but the scalar product is not always positive-defined and thenthe probabilistic interpretation is missing. Appendix B: The Noether Symmetry Approach
The Noether Symmetry Approach allows to find out sym-metry transformations and conserved quantities by means itis possible to reduce and solve dynamical systems. In thisappendix, we outline the main properties of the Noether Ap-proach and point out how to use it in Quantum Cosmology.Let L ( t, φ i , ˙ φ i ) be a non-degenerate Lagrangian describinga dynamical system. An infinitesimal transformation involv-ing the fields φ i and the coordinates is L ( t, φ i ˙ φ i ) → L ( t, φ i , ˙ φ i ) t = t + (cid:15)ξ ( t, φ i ) + O ( (cid:15) ) φ i = φ i + (cid:15)η i ( t, φ i ) + O ( (cid:15) ) . (B1)The generator of the transformation can be written as: X [1] = ξ ∂∂t + η i ∂∂φ i + ( ˙ η i − ˙ φ i ˙ ξ ) ∂∂ ˙ φ i . (B2)Noether’s first theorem states that if the relation X [1] L + ˙ ξ L = ˙ g (B3)3holds, then the quantity I ( t, φ i , ˙ φ i ) = ξ (cid:18) ˙ φ i ∂ L ∂ ˙ φ i − L (cid:19) − η i ∂ L ∂ ˙ φ i + g ( t, φ i ) , (B4)is a constant of motion. The function ξ depends on all the vari-ables of the configuration space and represents the infinitesi-mal generator related to the change of coordinates; η i is, inturn, related to the fields variation. The function g ( t, φ i ) is afour divergence whose value does not affect the dynamics ofthe system. This is the general version of Noether’s theoremholding both for internal and external symmetries. A partic-ular case can be obtained by the condition ξ = 0 . In the ap-plications to teleparallel Lagrangians discussed in this work,we only considered internal symmetries, where the Noethervector is written as: X = η i ∂∂φ i + ˙ η i ∂∂ ˙ φ i (B5)and the identity (B3) reduces to X L = 0 after imposing g = 0 . This means that, if the transformation does not involvecoordinates, internal symmetries arise if the Lie derivative ofthe Lagrangian along the flux of X vanishes: L X L = X L = η i ∂ L ∂φ i + ˙ η i ∂ L ∂ ˙ φ i (B6)and phase flux is conserved along X. Therefore, setting to zerothe Lie derivative of Lagrangian it is possible to find the sym-metries of the theory and the corresponding conserved quan-tity, that is Σ = η i ∂ L ∂ ˙ φ i . (B7)Notice that Eq. (B4) reduces to Eq. (B7) when ξ = 0 .In Quantum Cosmology, the configuration space is the min-isuperspace and, to find the Hamiltonian and to solve theWheeler-DeWitt equation, cyclic variables are identified byan appropriate change of coordinates. A possible transforma-tion is such that the conjugate momentum of the new variable z coincides with the conserved quantity Σ provided by theNoether symmetry, that is: π z = Σ . (B8) According to Eq. (B7), the above relation can be written as: (cid:18) η i ∂ L ∂ ˙ φ i (cid:19) = ∂ L ∂ ˙ z , (B9)In this way, also the generator of the symmetry must transformaccordingly; let X (cid:48) be the generator written in terms of thenew variables: X (cid:48) = η (cid:48) i ∂∂φ i + ˙ η (cid:48) i ∂∂ ˙ φ i (B10)Setting the component η (cid:48) of the new infinitesimal generator η (cid:48) i equal to 1 and η (cid:48) j (with j (cid:54) = 1 ) equal to zero, the conservedquantity (B7), corresponding to the variable φ , turns out to beequal to the conjugate momentum π φ . More formally, let usconsider the coordinates transformation φ i → Φ( φ i ) and thecorresponding inner derivative i X d Φ ≡ ˙ η (cid:48) i ∂ Φ ∂φ i . (B11)The new generator can be written in terms of the inner deriva-tive as: X (cid:48) = ( i X d Φ k ) ∂∂ Φ k + ddt ( i X d Φ k ) ∂∂ ˙Φ k (B12)so that, by setting the first component of the new infinitesimalgenerator equal to 1 and the others equal to zero, we obtain i X d Φ = η (cid:48) i ∂ Φ ∂φ i = 1 i X d Φ j = η (cid:48) i ∂ Φ j ∂ Φ i = 0 j (cid:54) = 1 . (B13)The conditions (B13) allows to introduce a cyclic variable bymeans of a methodical procedure. This procedure, after quan-tizing the theory and getting the Wave Function of the Uni-verse, points out when the Hartle criterion holds: conservedmomenta correspond to oscillating components of the wavefunction. In this case the Hamilton-Jacobi equations can beexactly integrated giving classical trajectories. According tothe Hartle criterion, such trajectories correspond to observableuniverses. 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